Differentiation126P1^
5
ACTIVITY 5.2 Take points X, Y, Z on the curve y = x^2 with x co-ordinates 3.1, 3.01 and 3.001
respectively, and find the gradients of the chords joining each of these points
to (3, 9).It looks as if the gradients are approaching the value 6, and if so this is the
gradient of the tangent at (3, 9).
Taking this method to its logical conclusion, you might try to calculate the
gradient of the ‘chord’ from (3, 9) to (3, 9), but this is undefined because there is a
zero in the denominator. So although you can find the gradient of a chord which
is as close as you like to the tangent, it can never be exactly that of the tangent.
What you need is a way of making that final step from a chord to a tangent.
The concept of a limit enables us to do this, as you will see in the next section. It
allows us to confirm that in the limit as point Q tends to point P(3, 9), the chord
QP tends to the tangent of the curve at P, and the gradient of QP tends to 6 (see
figure 5.5).The idea of a limit is central to calculus, which is sometimes described as the study
of limits.Historical note This method of using chords approaching the tangent at P to calculate the gradient
of the tangent was first described clearly by Pierre de Fermat (c.1608−65). He spent
his working life as a civil servant in Toulouse and produced an astonishing amount
of original mathematics in his spare time.Finding the gradient from first principles
Although the work in the previous section was more formal than the method of
drawing a tangent and measuring its gradient, it was still somewhat experimental.
The result that the gradient of y = x^2 at (3, 9) is 6 was a sensible conclusion,
rather than a proved fact.
In this section the method is formalised and extended.
Take the point P(3, 9) and another point Q close to (3, 9) on the curve y = x^2.
Let the x co-ordinate of Q be 3 + h where h is small. Since y = x^2 at Q, the
y co-ordinate of Q will be (3 + h)^2.P (3, 9)Q
Figure 5.5